Now that we have briefly seen the definitions and basic constructions, we can gain more insights into these structures and what they represent. First, let us understand why the lattice of subspaces (i.e. quantum logic) fails to be a Boolean algebra. Stone's representation theorem tells us that any Boolean algebra can be expressed as an algebra of sets using the standard set operations (i.e. intersection, union and complement) (cf. [1]}). For the lattice of subspaces, note that the disjunction \(p \vee q\)  does not correspond to the set union of the subspaces, but the span. In fact, the union of subspaces is not part of the lattice. This is why the lattice, even if it is a complemented \(0-1\)  lattice, fails to be a Boolean algebra: it does not use the standard set operations.
